Coherent integration loss due to white Gaussian phase noise

We develop a simple analytic expression for the change in coherent weighted integration gain due to a white Gaussian error or noise in the phase of the integrated samples. Our expression is shown by simulation to be very accurate for any reasonable value of phase noise standard deviation. The result is useful in estimating the performance impact on coherent signal processing systems of oscillator noise, residual motion compensation errors, and other system imperfections that are manifested primarily as phase errors.